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Definition of arctan(x) Functions

Examining the graph of tan(x), shown below, we note that it is not a one to one function on its implied domain. But if we limit the domain to \( ( -\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \), blue graph below, we obtain a one to one function that has an inverse which cannot be obtained algebraically.

The inverse function of \( f(x) = \tan(x) , x \in (-\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \) is \( f^{-1} = \arctan(x) \)
We define \( \arctan(x) \) as follows \[ y = \arctan(x) \iff x = \tan(y) \] where \( x \in (- \infty , +\infty) \) and \( y \in ( -\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \)

Let us make a table of values of \( y = \arctan(x) \) and graph it along with \( y = \tan(x), x \in ( -\infty , + \infty ) \)

\( x \)	-10000	-100	-10	-1	0	1	10	100	10000
\( y = \arctan(x) \)	-1.57069	-1.56079	-1.47112	\( -\dfrac{\pi}{4} \)	0	\( \dfrac{\pi}{4} \)	1.47112	1.56079	1.57069
	calculator	calculator	calculator	since \( \tan(-\dfrac{\pi}{4}) = - 1 \)	since \( \tan(0) = 0 \)	since \( \tan(\dfrac{\pi}{4}) = 1 \)	calculator	calculator	calculator

The graphs of \( y = \arctan(x) \) and \( y = \tan(x) , x \in ( -\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \) are shown below. Being inverse of each other, each of the two graphs is the reflection of the other on the line \( y = x \).
Note: Because \( \tan(x) \) has vertical asymptotes at \( x = - \dfrac{\pi}{2} \) and \( x = \dfrac{\pi}{2} \) (blue broken lines), its inverse function \( arctan(x) \) has horizontal asymtotes (red broken lines).
The behaviour of the graph of \( \arctan(x) \) close to the horizontal asymptotes may be described using the concept of limits as follows:
\( \lim_{x\to\infty} \arctan(x) = \dfrac{\pi}{2} \)
and
\( \lim_{x\to -\infty} \arctan(x) = - \dfrac{\pi}{2} \) Example 1
Evaluate \( arctan(x) \) given the value of \( x \).
Special values related to special angles
\( \arctan(0) = 0\) because \( \tan(0) = 0 \)
\( \arctan(-1) = -\dfrac{\pi}{4} \; \text{ or } -45^{o} \) because \( \tan(-\dfrac{\pi}{4}) = -1 \)
\( \arctan(1) = \dfrac{\pi}{4} \; \text{ or } 45^{o}\) because \( \tan(\dfrac{\pi}{4}) = 1 \)
Use of calculator
\( \arctan(-2) = -1.107 \; \text{ or } -63.43^{o} \)
\( \arctan(180) = 1.56 \; \text{ or } 89.68^{o} \)
\( \arctan(-230) = -1.566 \; \text{ or } -89.75^{o} \)
\( \arctan(-0.2) = -0.197 \; \text{ or } -11.31^{o} \)

Properties of \( y = arctan(x) \)

1. Domain: \( (-\infty , +\infty) \)
   
2. Range: \( (-\dfrac{\pi}{2} , \dfrac{\pi}{2}) \)
   
3. \( \arctan(-x) = - \arctan(x) \) , odd function
   
4. \( \arctan(x) \) is a one to one function
   
5. \( \tan(\arctan(x)) = x \) , due to property of a function and its inverse :\( f(f^{-1}(x) = x \) where \( x \) is in the domain of \( f^{-1} \)
   
6. \( \arctan(\tan(x)) = x \) , for x in the interval \( (-\dfrac{\pi}{2},\dfrac{\pi}{2}) \) , due to property of a function and its inverse :\( f^{-1}(f(x) = x \) where \( x \) is in the domain of \( f \)

Example 2
Find the range of the functions:
a) \( y = 3 \arctan(x)\)       b) \( y = - \arctan(x) + \pi/2 \)       c) \( y = 2 \arctan(x + 3) - \pi/4 \) Solution to Example 2
a)
the range is found by first writing the range of \( \arctan(x)\) as a double inequality
\( -\dfrac{\pi}{2} \lt \arctan(x) \lt \dfrac{\pi}{2} \)
multiply all terms of the above inequality by 3 and simplify
\( - 3 \pi / 2 \lt 3 \arctan(x) \lt 3 \pi / 2 \)
the range of the given function \( y = 3 \arctan(x) \) is given by the interval \( ( - 3 \pi / 2 , 3 \pi / 2 ) \).

b)
we start with the range of \( \arctan(x)\)
\( -\dfrac{\pi}{2} \lt \arctan(x) \lt \dfrac{\pi}{2} \)
multiply all terms of the above inequality by -1 and change symbol of the double inequality
\( -\dfrac{\pi}{2} \lt -\arctan(x) \lt \dfrac{\pi}{2} \)
add \( \dfrac{\pi}{2} \) to all terms of the double inequality above and simplify
\( 0 \lt - \arctan(x) + \dfrac{\pi}{2} \lt \pi \)
the range of the given function \( y = - \arctan(x) + \dfrac{\pi}{2} \) is given by the interval \( ( 0, \pi ) \).

c)
The graph of the function \( y = \arctan(x+3)\) is the graph of \( \arctan(x)\) shifted 3 unit to the left. Shifting a graph to the left or to the right does not affect the range. Hence the range of \( \arctan(x+3)\) is given by the double inequality
\( -\dfrac{\pi}{2} \lt \arctan(x+3) \lt \dfrac{\pi}{2} \)
Multiply all terms of the double inequality by 2 and simplify
\( - \pi \lt \arctan(x+3) \lt \pi \)
Add \( -\pi/4 \) to all terms of the above inequality and simplify to obtain the range of the given function \( y = 2 \arctan(x + 3) - \pi/4 \) by the double inequality
\( - 5\pi / 4 \lt \arctan(x+3) \lt 3 \pi / 4 \)

Example 3
Evaluate if possible
a) \( \tan(\arctan(-1.5))\)       b) \( \arctan(\tan(\dfrac{\pi}{7}) ) \)       c) \( \arctan(\tan(\dfrac{-\pi}{2})) \)       d) \( \tan(\arctan(-19.5)) \)       e) \( \arctan(\tan(\dfrac{ 13 \pi}{6}) ) \) Solution to Example 3
a)
\( \tan(\arctan(-1.5)) = -1.5 \) using property 5 above
b)
\( \arctan(\tan(\dfrac{\pi}{7}) = \dfrac{\pi}{7}\) using property 6 above
c)
NOTE that we cannot use property 6 because \( \tan(\dfrac{-\pi}{2}) \) is undefined
\( \arctan \tan(\dfrac{-\pi}{2}) \) is undefined
d)
\( \tan(\arctan(-19.5)) = -19.5\) use property 5
e)
Property 6 cannot be applied to the question in part e) because \( \dfrac{13\pi}{4} \) is not in the domain of that property. Hence we first calculate \( \tan(\dfrac{ 13 \pi}{4}) \)
\( \tan(\dfrac{ 13 \pi}{4}) = \tan(\dfrac{ 12 \pi}{4}+\dfrac{ \pi}{4}) = \tan(3\pi+\dfrac{ \pi}{4}) = \tan(\dfrac{ \pi}{4}) = 1 \)
We now substitute \( \tan(\dfrac{ 13\pi}{4}) \) by 1 in the given expression and simplify
\( \arctan(\tan(\dfrac{ 13 \pi}{6}) ) = \arctan(1) = \dfrac{\pi}{4} \)

Interactive Tutorial to Explore the Transformed arctan(x)

The exploration is carried out by analyzing the effects of the coefficients \( a, b, c\) and \( d \) included in the more general arctan function given by \[ f(x) = a \arctan(b x + c) + d \] Change coefficients \( a, b, c\) and \( d \) and click on the button 'draw' in the left panel below. Zoom in and out for better viewing. Depending on which coefficients are changing, we expect the graph to be transformed through vertical and horizontal compression, stretching, reflection and also vertical shifting.

1. Set the coefficients to \( a = 1, b = 1, c = 0 \) and \( d = 0 \) to obtain
   \[ f(x) = \arctan(x) \]
   Check that the domain of \( \arctan(x) \) is the set of all real numbers and the range is given by the interval \( (-\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \). Check also that \( \arctan(x) \) has horizontal asymptotes at \( y = -\dfrac{\pi}{2}\) and \( y = \dfrac{\pi}{2} \). (Zoom in and out)
2. Change coefficient \( a \) and note how the graph of \( a \arctan(x) \) changes (vertical compression, stretching, reflection). How does it affect the range and the asymptotes of the function?
   Does a change in coefficient \( a \) affect the domain of the function?
3. Change coefficient \( b \) and note how the graph of the function changes (horizontal compression, stretching). Does a change in \( b \) affect the range, domain and asymptotes of the function?
4. Change coefficient \( c \) and note how the graph of function changes (horizontal shift). Does a change \( c \) affect the domain, range and asymptotes of the function?
5. Change coefficient \( d \) and note how the graph of the function changes (vertical shift). Does a change in \( d \) affect the range, domain and asymptotes of the function?
6. If the range of \( \arctan(x) \) is given by the interval \( (-\dfrac{\pi}{2} , \dfrac{\pi}{2} ) \) what is the range of \( a \arctan(x) \)? What is the range of \( a \arctan(x) + d\)?
7. If the horizontal asymptotes of \( arctan(x) \) are given by the horizontal lines \( y = -\dfrac{\pi}{2} \) and \( y = \dfrac{\pi}{2} \) what are the horizontal asymptotes of \( a \arctan(x) \)?
8. What are the horizontal asymptotes of \( a \arctan(x) + d \)?

More References and Links to Inverse Trigonometric Functions

Inverse Trigonometric Functions
Graph, Domain and Range of Arcsin function
Graph, Domain and Range of Arctan function
Find Domain and Range of Arccosine Functions
Find Domain and Range of Arcsine Functions
Solve Inverse Trigonometric Functions Questions

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Source: https://www.analyzemath.com/Inverse-Trigonometric-Functions/Arctan.html

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